{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "The Fourier transform is perhaps the most fundamental tool in signal processing. Chapter 2 of [Müller, FMP, Springer 2015] approaches Fourier analysis from various perspectives and discusses their most important properties in the context of audio signal processing. In particular, the notion of a spectrogram, which yields a time–frequency representation of an audio signal, is introduced. \n", "
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"2.1 The Fourier Transform in a Nutshell
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"2.2 Signals and Signal Spaces
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"2.3 Fourier Transform
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"2.4 Discrete Fourier Transform (DFT)
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"2.5 Short-Time Fourier Transform (STFT)
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"2.6 Further Notes\n",
"
Topic | \n", "Relation to [Müller, FMP, Springer 2015] & Description | \n", "HTML | \n", "IPYNB | \n", "
Complex Numbers | \n", "[Section 2.2.3.1, Exercise 2.1] Absolute value; angle; polar representation; conjugation; inverse; arrow plot; polar coordinate plot | \n",
" [html] | \n", "[ipynb] | \n", "
Exponential Function | \n", "[Section 2.3.2.1, Exercise 2.15] Power series; Euler's formula; root of unity | \n",
" [html] | \n", "[ipynb] | \n", "
Discrete Fourier Transform (DFT) | \n", "[Section 2.1.3, Section 2.4] Inner product; DFT matrix; fast Fourier transform (FFT); runtime experiments | \n",
" [html] | \n", "[ipynb] | \n", "
DFT: Phase | \n", "[Section 2.3.2] Exponential function; polar coordinates; complex Fourier coefficient; phase; optimality property | \n",
" [html] | \n", "[ipynb] | \n", "
Discrete Short-Time Fourier Transform (STFT) | \n", "[Section 2.1.4, 2.5.3] Time localization; spectrogram; physical interpretation; interactive plot | \n",
" [html] | \n", "[ipynb] | \n", "
STFT: Influence of Window Function | \n", "[Section 2.5] Window type; window size | \n",
" [html] | \n", "[ipynb] | \n", "
STFT: Padding | \n", "[Section 2.1.4, Section 2.5.3] Padding variants; edge phenomena; interactive plot | \n",
" [html] | \n", "[ipynb] | \n", "
STFT: Frequency Grid Density | \n", "[Section 2.1.4] DFT frequency grid; zero padding; increasing density | \n",
" [html] | \n", "[ipynb] | \n", "
STFT: Frequency Interpolation | \n", "[Section 2.1.4, Section 3.1.1] Linear interpolation; cubic interpolation; log-frequency STFT; cents | \n",
" [html] | \n", "[ipynb] | \n", "
STFT: Inverse | \n", "[Section 2.4.2, Section 8.1.2.1] DFT; inverse DFT; window function; overlap–add technique; artition of unity | \n",
" [html] | \n", "[ipynb] | \n", "
STFT: Conventions and Implementations | \n", "[Section 2.1.4, 2.5.3] Time axis convention; centered windowing; frequency conversion; implementation issues | \n",
" [html] | \n", "[ipynb] | \n", "
Digital Signals: Sampling | \n", "[Section 2.2.2.1, Exercise 2.28] Equidistant sampling; aliasing; signal reconstruction; sinc function; sampling theorem | \n",
" [html] | \n", "[ipynb] | \n", "
Digital Signals: Quantization | \n", "[Section 2.2.2.2, Exercise 2.9] Uniform quantization; quantization error | \n",
" [html] | \n", "[ipynb] | \n", "
Interference and Beating | \n", "[Section 2.3.3.1, Exercise 2.19] Constructive interference; destructive interference; chirp; sweep | \n",
" [html] | \n", "[ipynb] | \n", "
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